Solid wall boundaries
Along solid wall or fully reflecting boundaries, the horizontal velocity
normal to the boundary must be zero over the entire water depth, i.e.,
u⋅ n
=
-h< z <η
(29)
0
where n is the normal vector to the boundary. This condition is satisfied in the
depth-integrated equations by specifying that both the volume flux density and
velocity normal to the boundary are zero, i.e.,
uα ⋅ n
=
(30)
0
uf ⋅ n =
0
(31)
Since the equations are solved on a staggered grid, the boundary conditions are
or vα = vf = 0 along wall boundaries perpendicular to the y-axis.
External wave generation boundaries
vα, and flux densities uf or vf corresponding to an incident storm condition are
specified. The time-histories may correspond to regular or irregular, unidirec-
tional or multidirectional waves.
Regular Waves. Regular, long-crested wave conditions are specified in
terms of a wave height, H, wave period, T, and direction of propagation, θ. The
water-surface elevation for small-amplitude waves (H << h) may be written as:
H
cos (kx cos θ + ky sin θ - ωt )
η( x, y, t )
=
(32)
2
where ω = 2π/T is the angular frequency, k is the wave number, and θ is the
direction of wave propagation relative to the positive x-axis. The boundary
conditions along a wave generation line perpendicular to the x-axis may be
obtained from the linearized form of the continuity equation (Equation 5) for
water of constant depth as:
=
Tu (ω) cos θ η( xg , yg , t )
uα ( xg , yg , t )
(33)
α2 1
=
(h + η) - h
- (kh) uα ( xg , yg , t )
2
(34)
2
u f ( xg , yg , t )
2 6
16
Chapter 3 Numerical Solution
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